We apply the traditional Dirac's bra-ket notations [1]} to denote generic quantum states: \(\langle \varphi |\in \mathcal {H}^*,|\varphi \rangle \in \mathcal {H}\) , where \(\mathcal {H}\)  represents a Hilbert space (over the reals \(\mathbb {R}\) ) with dimension \(d\) . For arbitrary \(|\varphi \rangle , |\psi \rangle \in \mathcal {H}\) , we denote their inner product as \(\langle \psi |\varphi \rangle \) . The norm of \(\varphi \)  is defined as \(\Vert \varphi \Vert  = \sqrt{\langle \varphi |\varphi \rangle }\) .
Let \(B(\mathcal {H})\)  be the space of all positive, Hermitian, and bounded operators from \(\mathcal {H}\)  to itself, with their norms defined as their largest absolute value of the eigenvalues \(\Vert \cdot \Vert _{1}\) . Let \(\mathcal {S}\)  be the subset of \(B(\mathcal {H})\)  such trace of its operators is 1. The space of all positive-operator valued measurements is denoted as \(\mathcal {V}\) .
